12 Days of Christmas GMAT Competition - Day 5: Amy and Beth, both work

(1) Amy alone can complete the same task in 8 days
This statement alone is insufficient because we don't know anything about Beth. Further, we also don't know anything about the total work.

(2) Beth alone can complete the same task in 20 days
This statement is insufficient because we don't know anything about Amy. Further, we also don't know anything about the total work.

Now combining 1 and 2
Let's assume total work to be 40 units
Amy per day efficiency= 5 units
Beth per day efficiency= 2 units
Now we know that they started working alternatively
If Amy starts then
Day 1 --> 5 units
Day 2 --> 2 units
Day 3 --> 5 units
Day 4 --> 2 units
Day 5 --> 5 units
Day 6 --> 2 units
Day 7 --> 5 units
Day 8 --> 2 units
Day 9 --> 5 units
Day 10 --> 2 units
Day 11--> 5 units
Work gets completed on 11th day itself.
If Beth starts
Day 1 --> 2 units
Day 2 --> 5 units
Day 3 --> 2 units
Day 4 --> 5 units
Day 5 --> 2 units
Day 6 --> 5 units
Day 7 --> 2 units
Day 8 --> 5 units
Day 9 --> 2 units
Day 10 --> 5 units
Day 11--> 2 units
Day 12--> 3 units
work gets completed on day 12
Therefore combining 1 and 2 is also insufficient
Option E

Given: Amy and Beth, both working at their respective constant rates, were assigned a task that they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed.
Asked: If they started the task from 1st of January, then on which day in January will the task get completed?


Since days in which Beth alone can complete the same task is unknown.
NOT SUFFICIENT


Since days in which Amy alone can complete the same task is unknown.
NOT SUFFICIENT

(1) + (2)



Case 1: Amy started on 1st of January
Task completion = 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8
Task completion requires 11 days and will be completed on 11th January.

Case 2: Beth started on 1st of January
Task completion = 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 3/40
Task completion requires 12 days and will be completed on 12th January.

NOT SUFFICIENT

IMO E

Condition 1:
We know Amy alone can complete within 8 days, but we don't know how long Beth can take. Then we can't find when the work will complete. Eliminate A and D.

Condition 2:
We know Beth alone can complete within 20 days, but we don't know how long Amy can take. Then we can't find when the work will complete. Eliminate B.

Condition 1 & 2:
We know amount of work both Amy and can do in a day. So,

Every 2 days, the amount of work that can complete is:
1/8 +1/20 = (5+2)/40 = 7/40


But still, we don't know who will start the work.
After 5x2= 10 days, the amount of work completed will be 5x7/40 = 35/40.
If Amy works on 11th day, then the total amount is 35/40+ 1/8 = 1. The work will be done on 11th January.
But, if Beth works on 11th day, then the total amount of work 35/40+1/20 = 37/40. The work will not complete 11th January, but will complete on 12th January.

So 2 answers are possible and we can't say when the work will complete. Eliminate option C.

Hence E is the best answer choice.

When we speak about two people doing some work together, we need to know the respective speeds of both of them - otherwise, there's no way of calculating the time they need to finish the task. That being said, conditions 1 and 2 will obviously not suffice separately (options A, B and D are out).

However, will we have enough information from both conditions combined?
Amy's speed will be \(a=\frac{1}{8}=\frac{5}{40}\), and Beth's will look as follows: \(b=\frac{1}{20}=\frac{2}{40}\)

Together, in any two consecutive days, they will manage to do \(a+b=\frac{5}{40} +\frac{ 2}{40} =\frac{ 7}{40}\) of their task. Therefore, after 10 days of work, they will finish \(5*\frac{7}{40} = \frac{35}{40}\).

And here's what the trouble is: now it becomes really important, who starts to work on January 1st. If it's Amy, then on January 11th she'll finish the job.
However, if it's Beth's turn, she will only do a small part, and the task will be finished only the following day by Amy, with \(\frac{35}{40}+\frac{2}{40}+\frac{5}{40} = \frac{42}{40}\).

Therefore, even two options together aren't sufficient, so the answer is E.

(1) Amy alone can complete the same task in 8 days
Cannot find the days required to complete the task as the rate of work of Beth is not available

(2) Beth alone can complete the same task in 20 days
Cannot find the days required to complete the task as the rate of work of Amy is not available

Combining (1) and (2) we get the following scenarios:

a ) Amy starts on the first day and ends on the last day
Let's suppose Beth takes 'n' days to complete, hence
=> (n+1)(1/8) +n(1/20) = 1
=> n(7/40) = 7/8
=> n = 5 days

a ) Amy starts on the first day and Beth ends on the last day
Let's suppose Beth takes 'n' days to complete, hence
=> n(1/8) +n(1/20) = 1
=> n(7/40) = 1
=> n = 40/7 ≈ 6 days

a ) Beth starts on the first day and ends on the last day
Let's suppose Amy takes 'n' days to complete, hence
=> n(1/8) + (n+1)(1/20) = 1
=> n(7/40) = 19/20
=> n = (19/20)
=> n = 38/7 ≈ 6 days

Hence we get different values for the times.

Option E